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I am just thinking about why this is true. Can I change it to

Q1. If a series is convergent then the series can be regrouped without changing the order of terms.

For example the sum of $(-1)^n$ is an alternating sequence and it is divergent, so I can't regroup them?

Q2. Can I claim that a convergent, non-alternating series be absolutely convergent?

As there is no difference after the term become absolute value, it should be still convergent after absolute those terms?

Q3. What does conditionally convergent mean? If a sequence is either convergent or absolutely convergent then it is conditionally convergent?

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Q2, Q3 are not related to the topic but it helps me think how these stuff work –  Timothy Leung May 31 '13 at 1:57

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Answers:

An absolutely convergent series can be summed in any fashion. The value of the sum is not affected by order. Your supposition about $(-1)^n$ is completely correct. Based on regrouping and rearranging, you could assign a wealth of values to the series with $(-1)^n$ as the summand. Even a conditionally convergent series can be rearranged to give any value.

A non-alternating, convergent series is also absolutely convergent since you can either factor out a negative (if the series is made of only negative terms) or leave it as is and it will be absolutely convergent. It does not follow, however, that any convergent series is also absolutely convergent. Case in point: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n} = \log(2)$, but the harmonic series diverges.

Conditionally convergent means that a series converges as is, but does not converge absolutely. Absolute convergence is much stronger than conditional convergence.

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